Fractal Theory

The fractal theory, introduced by Mandelbrot in 1982, studies patterns in the highly complex and unpredictable structures that exist in nature. Hutchinson in 1981, showed the construction of interesting fractals by iteration of maps in an Iterated Function System (IFS). The topological properties of an attractor are objects of great interest in the theory of dynamical systems because they help to understand the behavior of the orbits. It is often reasonable in dynamical systems theory to ignore any behavior which occurs only on a set of measure zero since such behavior will never be observed in any real world application. This suggests applying ergodic theory, which leads to an important connection between dynamical systems theory and ergodic theory.